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Detection of fluid motion direction based on the rotational Doppler effect of grafted perfect vortex beam

Zhenzhong Lu, Min Liu, Ziqi Hu, Biao Han, Yanling Sun, Jiali Liao, Zihao Wang, Shangtao Huang, and Pengyu Shi

Open Access Open Access

Abstract

Vortex beams have attracted much attention due to their unique rotational Doppler effect. With the in-depth study of vortex beams, many new vortex beams have been proposed gradually, while the detection of fluid motion is of great significance for the study of ocean turbulence. Based on the rotational Doppler effect of the grafted perfect vortex beam, we propose a non-embedded optical method for real-time detection of the magnitude and direction of fluid velocity and establish a two-dimensional fluid model for simulation verification. It is proved that the grafted perfect vortex beam can detect the magnitude and direction of the fluid velocity at the same time, which may provide a new way and theoretical support for the detection of fluid motion direction.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Turbulence is inevitable in the ocean, so the detection of turbulence in the ocean is very important for humans to better explore and utilize the ocean resources. Turbulence is a complex fluid phenomenon caused by irregular fluid density fluctuations that result in different refractive indices of the fluid. The understanding and study of turbulence have gone through a long and complex process and are still under study.

The initial methods used to study the flow field properties were mainly based on the thermal principle detection technique [1–3], but thermal technology has the disadvantage of contact measurement, which will cause a great interference to the measured flow field; Later, the acoustic Doppler effect was gradually applied to flow field detection, but the acoustic method had a large attenuation, resulting in a limited detection range [4]. After that, optical techniques were applied to flow field detection, and various optical methods such as laser Doppler velocimetry, laser-induced fluorescence method, and particle imaging velocimetry were proposed successively to achieve quantitative measurement of turbulent velocity fields [5–7]. As the special beam carries orbital angular momentum, the rotational Doppler effect of the vortex beam is gradually used in the detection of convective field velocity and vorticity [8,9].

The previous method of detecting the velocity of the flow field using the vortex beam can only detect the velocity magnitude, now we use grafted vortex beam to realize the simultaneous identification of velocity magnitude and direction of the flow field at the cross-section of the beam. In this paper, based on the rotational Doppler effect of the optical vortex, a detection method using grafted perfect vortex beam to detect the direction of fluid movement is proposed, and a simulation model is established to verify it.

2. Theoretical analysis

The vortex beam is also called spiral beam, its phase distribution is spiral. In the cylindrical coordinate system ($r,\theta ,z$), its complex amplitude expression contains the spiral phase term $\textrm{exp}({il\theta } )$, and l is the topological charge of the vortex beam, which theoretically can be taken as any finite value. The phase change of the beam around the optical axis is 2l$\mathrm{\pi }$. Figure 1 shows the intensity and phase distribution of the vortex beam with a topological charge of 10.

 figure: Fig. 1.

Fig. 1. Intensity and phase distribution of the vortex beam with the topological charge of 10. (a) intensity distribution, (b) phase distribution.

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Vortex beam is used in many fields because it carries orbital angular momentum. With its more and more extensive application, there is more and more demand for vortex beams. Researchers have made different forms of vortex beams according to their different needs [10–14]. The grafted vortex beam can be obtained by combining two or more vortex beams with different topological charges [15]. We propose to measure the flow field velocity and its direction using the grafted vortex beam.

As shown in Fig. 2(a), the vortex beam incidents perpendicular to the two-dimensional flow field, assuming that the velocity of the fluid has ${U_x}$ and $ {U_y}$ components in the x and y directions respectively, then the velocity of the fluid can be expressed as (${U_x}$, ${U_y}$). Consider a closed annular region of the radius of ${\rho _0}$, the angular component of the fluid velocity in polar coordinates is U (${\rho _0},\theta ).$

 figure: Fig. 2.

Fig. 2. Schematic diagram of grafted vortex beam detects the flow field. (a) grafted vortex beam incidents perpendicular to the flow field, (b) particles pass through the circular beam.

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The fluid system contains many independent tiny scatterers, as shown in Fig. 2(b), assuming that the scatterer passes through the annular incident light in the transverse position of $\vec{\rho }$, and the amplitude at that location can be expressed as:

$$E({\vec{\rho },t} )= {E_0}(\rho )\textrm{exp}\{{ - \textrm{i}[{2\mathrm{\pi }ft - \Phi ({\vec{\rho }} )} ]} \}$$
For the vortex beam, the phase $\mathrm{\Phi }({\vec{\rho }} )$ in its cross-section depends only on the azimuthal angle $\theta $ and the topological charge $l.$ When the moving scatterer passes through the ring beam, a phase gradient of ${\vec{\nabla }_ \bot }\mathrm{\Phi } = l\hat{\theta }/{\rho _0}$ is created. Since the scatterer moves laterally at velocity $\vec{U}$ within the beam plane, the echo signal is frequency shifted, expressed as [8]:
$$f({{\rho_0},\theta } )= \frac{l}{{2\mathrm{\pi }}} \cdot \frac{{U({{\rho_0},\theta } )}}{{{\rho _0}}}$$
For the grafted vortex beam carrying two topological charges l1 and l2, the frequency shifts generated by the echo signal in different spatial regions with annular spot topological charges l1 and l2 respectively are:
$${f_{{l_1}}}({{\rho_0},\theta } )= \frac{{{l_1}}}{{2\mathrm{\pi }}} \cdot \frac{{U({{\rho_0},\theta } )}}{{{\rho _0}}}\quad {f_{{l_2}}}({{\rho_0},\theta } )= \frac{{{l_2}}}{{2\mathrm{\pi }}} \cdot \frac{{U({{\rho_0},\theta } )}}{{{\rho _0}}}$$
From the equation, it can be seen that for grafted vortex beam, the Doppler shift generated at each point on the ring beam is not only related to the spot radius and azimuth angle, but also to the topological charge number of the region where the point is located. And from Eq. (2) and Eq. (3) we know that when the beam radius is known, the local velocity of the fluid can be calculated by both ordinary vortex or grafted vortex.

The Doppler frequency shift of each point on the ring beam is not necessarily the same, so the local average frequency shift $\left\langle f \right\rangle$ in the ring region is introduced as follows:

$$\left\langle f \right\rangle = \frac{1}{{2\mathrm{\pi }}} \cdot \int_0^{2\mathrm{\pi }} {f({{\rho_0},\theta } )} \textrm{d}\theta$$
Bringing Eq.  (2) into the above equation yields:
$$\left\langle f \right\rangle = \frac{1}{{4{\mathrm{\pi }^2}}} \cdot \int_0^{2\mathrm{\pi }} {{l_n}\frac{{U({{\rho_0},\theta } )}}{{{\rho _0}}}} \textrm{d}\theta$$
where n = 1,2 is associated with θ. Equation (5) is the average Doppler shift of the vortex beam. It can be seen that the average Doppler shift is related to the local velocity of the field and the topological charges l. Schematic diagrams of scattered particles in a fluid passing through an ordinary vortex beam and grafted vortex beam are shown in Fig. 3:

 figure: Fig. 3.

Fig. 3. Schematic diagram of scattered particles in the fluid passing through ordinary vortex beam and grafted vortex beam. (a) ordinary vortex beam, (b) grafted vortex beam.

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As can be seen from Fig. 3(a), for the ordinary vortex beam, when the direction of fluid motion and x-axis angle is $\theta $, at the azimuth of the annular spot for $\theta $ and $\theta + \pi $, the phase gradient direction is perpendicular to the direction of fluid motion, so the frequency shift of these two places is 0; while at the azimuth of the annular spot for $\theta + \frac{\pi }{2}$ and $\theta + \frac{{3\ast \pi }}{2}$, the phase gradient direction is the same and opposite respectively to the direction of fluid motion. Therefore, these two points correspond to two maximum frequency shifts, one positive and one negative with equal absolute values, respectively. That is, for the ordinary vortex beam, the scattering echo spectrum is symmetric along 0 frequency and does not change with the change of motion direction. In contrast, for the grafted vortex beam in Fig. 3(b), except for the case where the direction of fluid motion is perpendicular to the direction of the vortex beam being grafted, the topological charge distribution is different on both sides of the beam, so the resulting spectrum is no longer symmetric about the 0 frequency, and the maximum frequency shift changes as the direction of fluid motion changes.

Therefore, for an ordinary vortex beam, the average Doppler shift obtained after integrating the Doppler shift along the annular spot is zero; while for grafted vortex beam, bounded by the direction of fluid motion speed, the rotational Doppler shift symbols generated on both sides are of opposite magnitude and unequal size, and a non-zero value can be obtained after integration. As the direction of fluid motion changes, the topological charge distribution carried by the two halves of the ring beam separated by the direction of fluid motion also changes, which leads to a change in the local average Doppler shift obtained after integration. This paper is based on this principle to determine the direction of fluid motion by calculating the spectral integration of the grafted vortex beam scattering echoes.

3. Simulation model

Based on the principle of optical heterodyne detection, we use the grafted perfect vortex beam as the detection light to establish a flow field motion detection model. Since the ring radius on the left and right sides of the grafted vortex beam is different, to get grafted perfect vortex beam, as in Fig. 4, we first superimpose the axial cone phase on the grafted phase to convert the Laguerre-Gaussian beam into Bessel beam [16], then select the required diffraction order by superimposing the blazed grating of the appropriate period, and introduce a Fourier lens to obtain approximately perfect vortex beam at the focal plane of the lens finally.

 figure: Fig. 4.

Fig. 4. The generation process of grafted perfect vortex phase. (a) grafted vortex phase, (b) axial cone phase beam, (c) blazed grating phase, (d) grafted perfect vortex phase.

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The simulation model is divided into three parts: the incident system, the flow field system, and the receiving system. First, the incident system generates an ordinary Gaussian beam and a grafted perfect vortex beam with topological charge l1 and l2 and incident the grafted perfect vortex beam as the detection beam in the flow field, and the Gaussian beam as the reference beam without passing through the flow field; the flow field system generates a stable flow field with controllable velocity and direction of fluid particles, and loads the flow field motion information into the forward scattering signal of grafted perfect vortex beam; the receiving system interferes the scattered echo with Gaussian beam, and then Fourier transforms the interfered signal afterward, and finally obtains the Doppler shift corresponding to the velocity of the flow field motion. In this simulation model, assuming that there are M independent scattering particles moving at a certain speed in the transmission section of the beam, scattered echoes will be generated when the particles pass through the ring spot, and the echo signal can be obtained by superimposing the scattered echoes generated by each particle. The signal contains the position information of the moving scattering particles over time, thus the time domain signal can be obtained by sampling it for a certain period of time.

The grafted vortex beam incidents perpendicular to the fluid system as shown in Fig. 2(a), the moving fluid will generate scattered echoes passing the ring probe beam.

And the velocity of the fluid at each point on the ring observation area is not necessarily the same, so the rotational Doppler frequency shift generated at each point is not necessarily the same, i.e., the forward scattered signal received by the receiving system is a composite signal composed of different frequency signals. As a result, speckle noise is generated in the spectrum measurement, in addition to the random noise introduced in the model, which degrades the quality of the data. So we use the method of summing the spectra of multiple independent samples for spectrum accumulation to suppress the noise.

The number of accumulations is set to N = 300, and after accumulating the spectra, the rotational Doppler shift mean value can be calculated by calculating the weighted average of each frequency component in the spectrum.

Set the fluid to move in the two-dimensional plane. The direction of movement is the x-axis in Fig. 5(a), and the speed is constant at 2 mm/s. Set the topological charge of vortex beam l = 10 and the beam radius is 0.1 mm. After the Fourier transform of the sampled signal and subtraction of the initial frequency shift, the spectrum diagram shown in the corresponding position in Fig. 6(a) is obtained.

 figure: Fig. 5.

Fig. 5. Schematic diagram of fluid velocity direction when vortex beam with topological charge l is incident.

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 figure: Fig. 6.

Fig. 6. The scattering spectrum and average frequency shift of vortex beam with topological charge l = 10 incident on fluid. (a) spectrum diagram of different directions of fluid velocity when the vortex beams are incident, (b) the relationship between the average frequency shift and velocity direction.

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As can be seen in the spectrum corresponding to an angle of 0 degrees between the velocity direction and the x-axis in Fig. 6(a), there is a peak on both sides of zero frequency, and the peak value is the same. As can be seen from the spectrogram, the rotational Doppler frequency f = 31 Hz and the fluid velocity calculated according to Eq. (2) is 1.95 mm/s with an error of 2.5%.

Set the topological charge of vortex beams to be constant at 10 and the fluid velocity to be constant at 2 mm/s, and the vortex beam is rotated around the center of the beam to change the direction of movement of the fluid relative to the incident light. As shown in Fig. 5(b), the angle between the velocity direction and the x-axis in the figure is $\theta $, and the scattering spectrum and average frequency shift under different angles are obtained respectively in Fig. 6(a) and Fig. 6(b). In Fig. 6(a), the other three spectrums correspond to the angle between the direction of movement and the x-axis at an angle of 90°,180°and 270°, respectively.

It can be seen that when the fluid velocity is constant, the scattering spectrums are always the same, and the average frequency shift is always 0 regardless of the change in the fluid motion direction. In fact, the vortex beam is divided into two half rings with the fluid velocity direction as the dividing line. When the fluid velocity is constant, even if the velocity direction changes, the two half-loops will produce the positive and the negative rotational Doppler shift of equal absolute value. Therefore, the signal spectrum is symmetrical about the zero frequency. After the weighted average, the frequency shift is always 0. Therefore, the velocity of the fluid can be calculated by using the traditional vortex optical spectrum, but the direction of the velocity cannot be determined.

Changed the incident beam from the ordinary vortex beam to the grafted perfect vortex beam shown in Fig. 3(b). The topological charge on the left side of the beam is l1 and the topological charge on the right side is l2. Set the topological charge of the grafted vortex beam as l1 = 10, l2 = 20, and the beam radius as 0.1 mm. By rotating the grafted vortex beam to rotate the coordinate axis, the spectrogram when the motion direction and the x-axis angle is 0°, 90°, 180°, 270° is shown in Fig. 7(a). Keep the topological charge l1 constant and change the topological charge $l$2 to 30, 40, and 50, respectively, and repeat the previous operations to obtain the spectrogram shown in Fig. 7(b), (c), and (d).

 figure: Fig. 7.

Fig. 7. Spectrum diagram of different directions of fluid velocity when the grafted perfect vortex beams are incident. (a) topological charge l1 = 10 and l2 = 20, (b) topological charge l1 = 10 and l2 = 30, (c) topological charge l1 = 10 and l2 = 40, (d) topological charge l1 = 10 and l2 = 50.

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It can be seen from the figure that when the fluid velocity direction is in the x-axis direction, the rotational Doppler frequency shift generated by the upper and lower half rings with the x-axis as the boundary is equal in size and opposite in sign. Therefore, in this case, the spectrum is symmetrical about the zero frequency, and the weighted average of the spectrum results in 0; What's more, since the direction of particle motion is perpendicular to the grafting direction of vortex beam, at the junction of two vortex half-rings with different topological charges the direction of motion of the particles is the same and opposite to the direction of the phase gradient of the beam, respectively. Since the magnitude of the phase gradient is proportional to the topological charge of the vortex beam, the beam has two phase gradients of different magnitudes at the graft node of the grafted vortex beam. Therefore, two positive frequency shifts of different sizes are generated at the junction where the velocity direction is the same as the phase gradient direction, and two negative frequency shifts of different sizes are generated at the junction where the velocity direction is the opposite of the phase gradient, and four peaks appear in the spectrum. When the velocity direction of the fluid is the y-axis direction, the magnitude and sign of the rotational Doppler frequency shift produced by the left and right half rings with the y-axis as the boundary are different. The frequency shift produced by the side with large topological charge is larger than that produced by the side with small topological charge. In this case, the generated spectrum is asymmetric with respect to the zero frequency, and a positive value is obtained after the weighted average; On the contrary, when the velocity direction of the fluid is the negative direction of the y-axis, the weighted average of the frequency spectrum is negative.

Continuously change the velocity direction of the fluid, and calculate the weighted average value of the frequency spectrum of the signal when the fluid velocity is in different directions. Figure 8 shows the relationship between normalized frequency integration and velocity direction.

 figure: Fig. 8.

Fig. 8. Relationship between the Normalized frequency integration and velocity direction when grafted perfect vortex beam with topological charges of l1 = 10 and l2 = 20 is incident.

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It can be seen from the curve formed by the red spheres in Fig. 8 that the frequency integration (also called as average frequency shift) obtained by taking the grafted perfect vortex beam as the incident light and after the weighted average of the obtained frequency spectrum conforms to the sinusoidal distribution with the change of fluid velocity direction. When the angle between the velocity direction and the x-axis is 90° (y-axis positive direction), the circulation integral is the largest, and when the angle between the velocity direction and the x-axis is 270° (y-axis negative direction), the circulation integral is the smallest.

When the angle between the direction of motion and the x-axis is 0°and 180°, the value of the frequency integral is both 0. However, when the angle is 0°, the same phase gradient direction as the direction of motion exists in the half-loop with large topological charge due to the gradual increase of the angle theta near the positive direction of the x-axis. Therefore, the absolute value of the positive frequency shift is larger than the absolute value of the negative frequency shift, resulting in the frequency integral increasing from 0 to positive value with increasing angle near the positive direction of the x-axis, i.e., the slope of the frequency integral curve is positive when the angle is 0°; on the contrary, the slope of the frequency curve is negative when the angle is 180°. In summary, the weighted average frequency shift (frequency integration) corresponds one-to-one to the direction of fluid motion, and the frequency integration obtained by grafted vortex beam can accurately identify the direction of flow field motion.

4. Conclusion

Based on the rotational Doppler effect of vortex beam, a method for detecting the direction of fluid motion by grafted perfect vortex beam is proposed in this paper. On this basis, a two-dimensional fluid model is established, and the ordinary vortex beam and the grafted perfect vortex beam are used as incident light for simulation. The simulation results show that the ordinary vortex beam can only detect the fluid velocity magnitude, while the grafted perfect vortex can also identify the fluid velocity direction. Which provides a new idea and theoretical basis for non-embedded detection of fluid movement direction.

Funding

Fundamental Research Funds for the Central Universities (ZYTS23128).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. Intensity and phase distribution of the vortex beam with the topological charge of 10. (a) intensity distribution, (b) phase distribution.

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Fig. 2.
Fig. 2. Schematic diagram of grafted vortex beam detects the flow field. (a) grafted vortex beam incidents perpendicular to the flow field, (b) particles pass through the circular beam.

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Fig. 3.
Fig. 3. Schematic diagram of scattered particles in the fluid passing through ordinary vortex beam and grafted vortex beam. (a) ordinary vortex beam, (b) grafted vortex beam.

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Fig. 4.
Fig. 4. The generation process of grafted perfect vortex phase. (a) grafted vortex phase, (b) axial cone phase beam, (c) blazed grating phase, (d) grafted perfect vortex phase.

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Fig. 5.
Fig. 5. Schematic diagram of fluid velocity direction when vortex beam with topological charge l is incident.

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Fig. 6.
Fig. 6. The scattering spectrum and average frequency shift of vortex beam with topological charge l = 10 incident on fluid. (a) spectrum diagram of different directions of fluid velocity when the vortex beams are incident, (b) the relationship between the average frequency shift and velocity direction.

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Fig. 7.
Fig. 7. Spectrum diagram of different directions of fluid velocity when the grafted perfect vortex beams are incident. (a) topological charge l1 = 10 and l2 = 20, (b) topological charge l1 = 10 and l2 = 30, (c) topological charge l1 = 10 and l2 = 40, (d) topological charge l1 = 10 and l2 = 50.

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Fig. 8.
Fig. 8. Relationship between the Normalized frequency integration and velocity direction when grafted perfect vortex beam with topological charges of l1 = 10 and l2 = 20 is incident.

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

E ( ρ → , t ) = E 0 ( ρ ) exp { − i [ 2 π f t − Φ ( ρ → ) ] }
f ( ρ 0 , θ ) = l 2 π ⋅ U ( ρ 0 , θ ) ρ 0
f l 1 ( ρ 0 , θ ) = l 1 2 π ⋅ U ( ρ 0 , θ ) ρ 0 f l 2 ( ρ 0 , θ ) = l 2 2 π ⋅ U ( ρ 0 , θ ) ρ 0
⟨ f ⟩ = 1 2 π ⋅ ∫ 0 2 π f ( ρ 0 , θ ) d θ
⟨ f ⟩ = 1 4 π 2 ⋅ ∫ 0 2 π l n U ( ρ 0 , θ ) ρ 0 d θ
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